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Unformatted text preview: SIEO 3600 (IEOR Majors) Solution to Assignment #4 Introduction to Probability and Statistics February 17, 2010 Solution to Assignment #4 1. A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die which is less than that appearing on the red die. (That is, if B ( R ) [ Y ] is the number appearing on the blue (red) [yellow] die, then we are interested in P ( { B < Y < R } ).) (a) What is the probability that no two of the dice land on the same number? (b) Given that no two of the dice land on the same number, what is the conditional proba- bility that { B < Y < R } ? (c) What is P ( { B < Y < R } )? (d) If we regard the outcome of the experiment as the vector B,R,Y , how many outcomes are there in the sample space? (e) Without using the answer to (c), determine the number of outcomes that result in event { B < Y < R } . (f) Use the results of parts (d) and (e) to verify your answer to part (c). Solution: (a) P ( { B,Y,R are different } ) = P 3 6 6 3 = 5 / 9. (b) By symmetry, P ( { B < Y < R }|{ B,Y,R are different } ) = 1 / 6. (c) Using the definition of conditional probability, P ( { B < Y < R } ) = P ( { B < Y < R }|{ B,Y,R are different } ) · P ( { B,Y,R are different } ) = (5 / 9)(1 / 6) = 5 / 54. (d) Total number of outcomes of the sample space is | S | = 6 3 = 216. (e) |{ B < Y < R }| = ( 6 3 ) = 20. Note, all you have to do is to choose 3 different numbers between 1 and 6. Say you get 1,2 and 5, then you have to assign 5 to R , 2 to Y and 1 to B to maintain their ordering....
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